Problem 9 - Fundamentals of Mathematics

Question

For problems 9-11, reduce, if possible, each fraction to lowest terms. (Section 4.4) \(\frac{21}{35}\)

Step-by-Step Solution

Answer

The fraction \( \frac{21}{35} \) simplifies to \( \frac{3}{5} \).

Step 1: Identify the GCD

To simplify the fraction \( \frac{21}{35} \), we first need to determine the greatest common divisor (GCD) of the numerator and the denominator. The numbers are 21 and 35. We will list the factors of each number to find the GCD.- Factors of 21: 1, 3, 7, 21- Factors of 35: 1, 5, 7, 35The greatest common factor is 7.

Step 2: Divide by the GCD

Now that we have the GCD, we can simplify the fraction by dividing both the numerator and the denominator by the GCD.1. Divide the numerator: \( \frac{21}{7} = 3 \)2. Divide the denominator: \( \frac{35}{7} = 5 \)Thus, the simplified fraction is \( \frac{3}{5} \).

Step 3: Verify the Simplification

To ensure that \( \frac{3}{5} \) is in its simplest form, check that the only common factor of the numerator (3) and the denominator (5) is 1. Since there are no common factors other than 1, the fraction is already in its lowest terms.

Key Concepts

Greatest Common Divisor

The greatest common divisor, or GCD, is an essential concept when it comes to simplifying fractions. The GCD of two numbers is the largest number that evenly divides both numbers. Finding the GCD is the first step in reducing a fraction to its simplest form.
Why is it important? Because knowing the GCD helps us understand how much we can "shrink" or simplify a fraction without changing its value. By dividing both the numerator (the top part of the fraction) and the denominator (the bottom part) by the GCD, we get the fraction in its reduced form. There are various methods to find the GCD, but listing all factors and identifying the largest common one is a straightforward approach. For example, if you have the numbers 21 and 35, you would list their factors and see that 7 is the largest number they both share. Thus, 7 is the GCD.

Factors of Numbers

Understanding the factors of numbers is a crucial step in learning how to simplify fractions. A factor of a number is any integer that divides that number without leaving a remainder. To find all the factors of a number, you can start by writing down 1 and the number itself, and then testing the integers between them. Knowing how to find these factors allows you to determine the common factors between two numbers, which is vital when you need to reduce fractions. For instance, if you have the numbers 21 and 35:

• Factors of 21: 1, 3, 7, 21
• Factors of 35: 1, 5, 7, 35

The factor 7 appears in both lists, making it the greatest common divisor. Learning to efficiently identify and list factors helps in quickly spotting the GCD, thereby simplifying the process of fraction reduction.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where there are no common factors between the numerator and the denominator other than 1. It's like squeezing a fraction into its basic building blocks without changing its overall value.To do this, you divide both the numerator and the denominator by their GCD. Using the example of \(\frac{21}{35}\), you found the GCD to be 7:

• Divide the numerator: \(\frac{21}{7} = 3\)
• Divide the denominator: \(\frac{35}{7} = 5\)

Thus, the fraction simplifies to \(\frac{3}{5}\).After simplifying, it's crucial to verify the result. Ensure that the only common factor between the new numerator and denominator is 1. If so, the fraction is in its simplest form. This process not only improves your arithmetic skills but also ensures accuracy in math tasks.